How can I delete text messages from my iPhone to clear up space?

I have a 16GB iPhone 5 and I keep getting messages saying that my phone is full and out of memory. Under the explanation of usage in the Settings I've found that I have 5.5GB of text messages. I have tried the following to remove them and still 5.5GB of text messages remain on my phone.

1. Delete the texts from the messages app.

2. Delete the texts from the messages app and restart the phone.

3. Delete the texts from the messages app, back up the phone, erase the phone, and restore the phone.

None of these solutions have worked to remove the messages. Can anyone please tell me how I can permanently delete messages to create more space on the hard drive?